# Proton Quantum Tunneling: Influence and Relevance to Acidosis-Induced Cardiac Arrhythmias/Cardiac Arrest

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## Abstract

**:**

## 1. Introduction

^{+}/K

^{+}ATPase is indirectly inhibited because acidosis inhibits the cellular metabolism and hence production of ATP, but this seems unlikely to happen because first acidosis does not inhibit the metabolism rigorously, and secondly because changes in membrane potential were noticed during times of normal intracellular ATP levels [4]. The second mechanism states that resting intracellular Ca

^{+2}concentration is increased, because of this, Na

^{+}/Ca

^{+2}exchanger may be activated or non-specific intracellular cation currents might be generated, but the fact that the currents generated are decreased by the rising intracellular Na

^{+}concentrations that happens during acidosis weakens this theory [4]. The third mechanism states that there is a decrease in K

^{+}currents during times of acidosis, but this mechanism is disproved somehow because firstly, an increase in intracellular Ca

^{+2}during acidosis as mentioned earlier activates calcium activated potassium channels, thus hyper-polarizing the membrane; secondly, increased intracellular sodium concentrations during acidosis might activate sodium activated potassium channels, therefore hyperpolarizing the membrane potential [4]. The fourth mechanism states that potassium piles up in the intercellular clefts of purkinje fibers during acidosis by inhibiting Na

^{+}/K

^{+}ATPase directly, but the gap is that this is difficult to be achieved by merely inhibiting the Na

^{+}/K

^{+}ATPase alone [4]. (2) Early afterdepolarization: acidosis generates early after depolarizations, which are produced by recovery of the inactivated L-type calcium channels during the repolarization phase of action potential, but the gap here is that acidosis inhibits calcium current directly and indirectly by increasing the intracellular calcium concentration [4]. (3) Delayed after depolarization: these are caused by increased intracellular Ca

^{+2}which provokes inward depolarizing currents mainly by activating the Na

^{+}/Ca

^{+2}exchanger, and this mechanism is opposed by the fact that the concentration of intracellular sodium ions increases during acidosis and this inhibits the currents mediated by Na

^{+}/Ca

^{+2}exchanger [4].

## 2. The Mathematical Model

_{Q}is the tunneling probability, m is the mass of the particle (Kg), $\hslash $ is the reduced Planck constant ($1.05\times {10}^{-34}$ Js), U(x) is the energy of the barrier with respect to the position of particle x, KE is the kinetic energy of the particle, and x2–x1 is the region where the energy of barrier is higher than the energy of the particle.

_{m}is the membrane potential, K

_{B}is the Boltzmann constant ($1.38\times {10}^{-23}$ J/K), T is the body temperature (310 K). Our model will be applied on protons and sodium ions, and both have charge equal to the charge of electron $1.6\times {10}^{-19}C$. The mass of proton is $1.67\times {10}^{-27}$ kg and the mass of sodium ion is $3.8\times {10}^{-26}$ kg.

_{Q}is the tunneling probability. The unit of quantum conductance of single channel is Siemens (S).

^{2}), and ${C}_{Q}$ is the quantum conductance of a single channel with the unit of (mS). Thus, the unit of quantum membrane conductance is mS/cm

^{2}.

_{m}is resting membrane potential.

^{2}[2,8], and $M{C}_{K}=0.5$ mS/cm

^{2}[2,8].

## 3. Results

_{m}is chosen to be the resting membrane potential ${V}_{m}=0.087$ V which represents the original and initial state of potential. As a result, ${E}_{Gate}=3.31$ J and can be 4.35 J if the gating charge is 5e [25]. On the other hand, this estimation cannot be used for the inactivation gate because the increase in half inactivation voltage should cause the energy of the gate to decrease since more energy is needed to inactivate the channel in this case. Thus, using this estimation will increase the energy of the inactivation gate instead of decreasing it. In other words, this equation ${q}_{gating}({V}_{1/2}-{V}_{m})$ can estimate the energy to inactivate the channel but not the gate’s energy ${E}_{Gate}$ that impedes the ions passage. Because these values are based on estimation, we will take a range of values for ${E}_{Gate}$ to include a wide range of possibilities for activation and inactivation gates. Additionally, the ranges will be chosen so that the substitution will not result in negative numbers in the square root in the equations of quantum tunneling (avoiding obtaining imaginary numbers). In our study, we assume that the energy of gate ${E}_{Gate}$ for both activation and inactivation gates is the same to simplify the mathematical evaluation of the effect of ${E}_{Gate}$ on quantum tunneling probability, quantum conductance of single channel, and quantum membrane conductance.

^{2}[8].

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## 4. Discussion

_{m}. See Figure 12.

_{m}= 0.087 V, V

_{m}= 0.077 V, and V

_{m}= 0.067 V). The evaluation is made by setting $L=1.5$ m and n = 1. See Table 2.

_{m}= 0.087 V, V

_{m}= 0.077 V, and V

_{m}= 0.067 V). The evaluation is made by setting $L=1.5$ m and n = 1. See Table 6.

- The tunneling probability of extracellular ions (protons and sodium ions) is higher than the tunneling probability of intracellular ions (protons and sodium ions) at the same setting values;
- The tunneling probability of extracellular and intracellular protons is higher than the extracellular and intracellular sodium ions at the same setting values;
- As the energy of the gate increases, the tunneling probability of ions (protons and sodium ions) decreases;
- As the gate length increases, the tunneling probability of ions (protons and sodium ions) decreases;
- As the absolute value of membrane potential (negative inside with regard to outside) increases, the tunneling probability of extracellular ions increases;
- As the location of gate (n) increases, the value of membrane potential available for the kinetic energy of extracellular ions decreases and thus their tunneling probability decreases.

_{m}= 0.087 V, V

_{m}= 0.077 V, and V

_{m}= 0.067 V). The evaluation is made by setting $L=1.5$ m and n = 1. See Table 10.

_{m}= 0.087 V, V

_{m}= 0.077 V, and V

_{m}= 0.067 V). The evaluation is made by setting $L=1.5$ m and n = 1. See Table 14.

^{2}, and n = 1. See Table 17.

_{m}= 0.087 V, V

_{m}= 0.077 V, and V

_{m}= 0.067 V). The evaluation is made by setting $L=1.5$ m, $D={10}^{11}$ channels/cm

^{2}, and n = 1. See Table 18.

^{2}. See Table 19.

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}). The evaluation is made by setting ${V}_{m}=0.087$ V, $L=1.5$ m, and n = 1. See Table 20.

^{2}. See Table 21.

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}). The evaluation is made by setting $L=1.5$ m. See Table 22.

^{2}at $D={10}^{11}$ channels/cm

^{2}, $4.36\times {10}^{4}$ mS/cm

^{2}at $D={10}^{10}$ channels/cm

^{2}, and $4.36\times {10}^{3}$ mS/cm

^{2}at $D={10}^{9}$ channels/cm

^{2}. By comparing these values with the values in tables of protons, it is clear that both extracellular protons at 2.5 J and intracellular protons at 1 J obtain quantum membrane conductance higher than the classical membrane conductance. This is true for all the values at 2.5 and 1 J. These higher quantum membrane conductance values continue to decrease until reaching at ${E}_{Gate}=7$ J. The classical membrane conductance of protons mediated by voltage-gated sodium channels is valid only when the channels are open and the open voltage-gated sodium channels are not available at the resting state, during repolarization, or after repolarization because voltage-gated sodium channels in these stages are either closed or inactivated. On the other hand, the quantum membrane conductance of protons is valid when the channels are either closed or inactivated, which are prominently available during the resting state, repolarization phase, and after repolarization. Interestingly, it is clear from the tables that protons achieve quantum membrane conductance higher than the leaky classical membrane conductance of potassium and sodium ions (0.5 mS/cm

^{2}and 0.005 mS/cm

^{2}, respectively) at the resting state. This predicts the ability of protons to depolarize the resting membrane potential via quantum tunneling since extracellular protons have higher quantum conductance than the intracellular protons and the intracellular to extracellular concentration ratio of protons, which is 10, is lower than the ratio of potassium ions, which is 140/4 = 35.

^{2}, and n = 1. See Table 23.

_{m}= 0.087 V, V

_{m}= 0.077 V, and V

_{m}= 0.067 V). The evaluation is made by setting $L=1.5$ m, $D={10}^{11}$ channels/cm

^{2}, and n = 1. See Table 24.

^{2}. See Table 25.

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}). The evaluation is made by setting ${V}_{m}=0.087$ V, $L=1.5$ m, and n = 1. See Table 26.

^{2}. See Table 27.

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}). The evaluation is made by setting $L=1.5$ m. See Table 28.

^{2}at $D={10}^{11}$ channels/cm

^{2}, $173$ mS/cm

^{2}at $D={10}^{10}$ channels/cm

^{2}, and $17.3$ mS/cm

^{2}at $D={10}^{9}$ channels/cm

^{2}. By comparing these values with the values in tables of sodium ions, it is clear that both extracellular sodium ions at 2.5 J and intracellular sodium ions at 1 J can obtain quantum membrane conductance higher than the classical membrane conductance. This is not true for all the values at 2.5 and 1 J because other factors such as gate length, membrane potential, and the gate location modulate the values of quantum membrane conductance. These high quantum membrane conductance values continue to decrease until reaching at ${E}_{Gate}=7$ J. The classical membrane conductance of sodium ions mediated by voltage-gated sodium channels is valid only when the channels are open and the open voltage-gated sodium channels are not available at the resting state, during repolarization, or after repolarization because voltage-gated sodium channels in these stages are either closed or inactivated. On the other hand, the quantum membrane conductance of sodium ions is valid when the channels are either closed or inactivated, which are prominently available during the resting state, repolarization phase, and after repolarization. Interestingly, it is clear from the tables that sodium ions achieve quantum membrane conductance higher than the leaky classical membrane conductance of potassium and sodium ions (0.5 mS/cm

^{2}and 0.005 mS/cm

^{2}, respectively) at the resting state. This predicts the ability of sodium ions to depolarize the resting membrane potential via quantum tunneling, since extracellular sodium ions have higher quantum conductance than the intracellular sodium ions and the extracellular concentration of sodium ions is higher than their intracellular concentration.

^{2}, and n = 1. See Table 29.

^{2}, and n = 1. See Table 30.

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}) and by setting $p{H}_{E}=7.4$, $L=1.5$ m, and n = 1. See Table 31.

^{2}. See Table 32.

^{2}, and n = 1. See Table 33.

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}) and by setting $L=1.5$ m, and n = 1. See Table 34.

^{2}. See Table 35.

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}). The evaluation is made by setting the permeability ratio ${P}_{H}/{P}_{Na}=252$ and the classical single channel conductance of sodium channel ${C}_{Na}=17.3\times {10}^{-12}$ S. See Table 36.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Represents the different locations of the gate through which quantum tunneling of ions occur. n = 1 is where the ion will pass through the entire membrane potential, n = 2 is where the ion will pass the half of membrane potential, and n = 4 is where the ion will pass the quarter of membrane potential.

**Figure 2.**(

**a**–

**c**): represents the mathematical graph of common logarithm of tunneling probability for extracellular protons ${\mathrm{log}}_{10}({T}_{Q})-{H}_{E}$ over gate’s energy range from 2.5 to 7 J according to gate length, membrane potential, and gate location, respectively; (

**d**): represents the mathematical graph of common logarithm of tunneling probability for intracellular protons ${\mathrm{log}}_{10}({T}_{Q})-{H}_{I}$ over gate’s energy range from 1 to 7 J according to gate length.

**Figure 3.**(

**a**–

**c**): represents the mathematical graph of common logarithm of tunneling probability for extracellular sodium ions ${\mathrm{log}}_{10}({T}_{Q})-N{a}_{E}$ over gate’s energy range from 2.5 to 7 J according to gate length, membrane potential, and gate location, respectively; (

**d**): represents the mathematical graph of common logarithm of tunneling probability for intracellular sodium ions ${\mathrm{log}}_{10}({T}_{Q})-N{a}_{I}$ over gate’s energy range from 1 to 7 J according to gate length.

**Figure 4.**(

**a**–

**c**): represents the mathematical graph of common logarithm of quantum conductance of single channel for extracellular protons ${\mathrm{log}}_{10}({C}_{Q})-{H}_{E}$ over gate’s energy range from 2.5 to 7 J according to gate length, membrane potential, and gate location, respectively; (

**d**): represents the mathematical graph of common logarithm of quantum conductance of single channel for intracellular protons ${\mathrm{log}}_{10}({C}_{Q})-{H}_{I}$ over gate’s energy range from 1 to 7 J according to gate length.

**Figure 5.**(

**a**–

**c**): represents the mathematical graph of common logarithm of quantum conductance of single channel for extracellular sodium ions ${\mathrm{log}}_{10}({C}_{Q})-N{a}_{E}$ over gate’s energy range from 2.5 to 7 J according to gate length, membrane potential, and gate location, respectively; (

**d**): represents the mathematical graph of common logarithm of quantum conductance of single channel for intracellular sodium ions ${\mathrm{log}}_{10}({C}_{Q})-N{a}_{I}$ over gate’s energy range from 1 to 7 J according to gate length.

**Figure 6.**(

**a**–

**d**): represents the mathematical graph of common logarithm of quantum membrane conductance for extracellular protons ${\mathrm{log}}_{10}(M{C}_{Q})-{H}_{E}$ over gate’s energy range from 2.5 to 7 J according to gate length, membrane potential, gate location, and channels density, respectively; (

**e**,

**f**): represents the mathematical graph of common logarithm of quantum membrane conductance for intracellular protons ${\mathrm{log}}_{10}(M{C}_{Q})-{H}_{I}$ over gate’s energy range from 1 to 7 J according to gate length, and channels density, respectively.

**Figure 7.**(

**a**–

**d**): represents the mathematical graph of common logarithm of quantum membrane conductance for extracellular sodium ions ${\mathrm{log}}_{10}(M{C}_{Q})-N{a}_{E}$ over gate’s energy range from 2.5 to 7 J according to gate length, membrane potential, gate location, and channels density, respectively; (

**e**,

**f**): represents the mathematical graph of common logarithm of quantum membrane conductance for intracellular sodium ions ${\mathrm{log}}_{10}(M{C}_{Q})-N{a}_{I}$ over gate’s energy range from 1 to 7 J according to gate length and channels density, respectively.

**Figure 8.**The relationship between the resting membrane potential and the energy of gate under the influence of quantum tunneling of protons according to external pH, gate length, channels density, and gate location.

**Figure 9.**The relationship between the resting membrane potential and the energy of gate under the influence of quantum tunneling of sodium ions according to gate length, channel density, and gate location.

**Figure 10.**The relationship between the resting membrane potential and a range of external pH from 5 to 7.4 under the influence of classical transport of protons through open voltage-gated sodium channels and at different channels densities.

**Figure 11.**A schematic diagram that represents the quantum tunneling of the wave-function of a proton through different levels of gate energy E3 > E2 > E1. The lower is the gate energy; the higher is the tunneling probability, which is represented by higher amplitude of wave-function after tunneling through the gate (shown in red).

**Figure 12.**A schematic diagram that represents the quantum tunneling of the wavefunction of extracellular and intracellular ions through the gate (red in color). (

**a**): extracellular ion has higher kinetic energy manifested as shorter wavelength, and higher tunneling probability manifested as higher amplitude after passing the gate; (

**b**): intracellular ion has lower kinetic energy manifested as longer wavelength and lower tunneling probability manifested as lower amplitude after passing the gate.

**Figure 13.**A schematic diagram that represents the quantum tunneling of extracellular proton and sodium ion. (

**a**): proton has longer wavelength (due to small mass) and higher tunneling probability manifested as higher amplitude after passing the gate; (

**b**): sodium ion has shorter wavelength (due to larger mass) and lower tunneling probability manifested as lower amplitude after passing the gate.

**Figure 14.**(

**a**): represents normal heart with normal polarization (negative inside with regard to outside); (

**b**): represents inward quantum tunneling of protons, which is indicated by inward arrows. This inward tunneling is responsible for membrane depolarization during acidosis according to the quantum model; (

**c**): represents the state of membrane depolarization (positive inside with regard to outside), which is the outcome of protons tunneling that increases the tendency of arrhythmias and cardiac arrest.

**Table 1.**Represents the values of quantum tunneling probability of extracellular protons that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m and by setting ${V}_{m}=0.087$ V and n = 1.

The Gate Length L (m) | ${\mathit{T}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{T}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | 0.24 | $2.6\times {10}^{-8}$ |

2 | 0.15 | $9.3\times {10}^{-11}$ |

2.5 | 0.094 | $2.9\times {10}^{-13}$ |

**Table 2.**Represents the values of quantum tunneling probability of extracellular protons that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at ${V}_{m}=0.087$ V, ${V}_{m}=0.077$ V and ${V}_{m}=0.067$ V and by setting L = 1.5 m and n = 1.

The Membrane Potential V_{m} (V) | ${\mathit{T}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{T}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

0.087 | 0.24 | $2.6\times {10}^{-8}$ |

0.077 | 0.11 | $1.1\times {10}^{-8}$ |

0.067 | 0.045 | $4.7\times {10}^{-9}$ |

**Table 3.**Represents the values of quantum tunneling probability of extracellular protons that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at n = 1, n = 2, and n = 4 and by setting ${V}_{m}=0.087$ V and $L=1.5$ m.

The Location of Gate n | ${\mathit{T}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{T}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1 | 0.24 | $2.6\times {10}^{-8}$ |

2 | $4\times {10}^{-3}$ | $6.9\times {10}^{-10}$ |

4 | $2.8\times {10}^{-4}$ | $8.3\times {10}^{-11}$ |

**Table 4.**Represents the values of quantum tunneling probability of intracellular protons that take the range between the two values calculated at ${E}_{Gate}=1$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m.

The Gate Length L (m) | ${\mathit{T}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=1\mathbf{J}$ | ${\mathit{T}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | 0.092 | $1.1\times {10}^{-11}$ |

2 | 0.042 | $2.4\times {10}^{-15}$ |

2.5 | 0.019 | $5.1\times {10}^{-19}$ |

**Table 5.**Represents the values of quantum tunneling probability of extracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m and by setting ${V}_{m}=0.087$ V and n = 1.

The Gate Length L (m) | ${\mathit{T}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{T}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | $1.2\times {10}^{-3}$ | $8.1\times {10}^{-37}$ |

2 | $1.2\times {10}^{-4}$ | $7.6\times {10}^{-49}$ |

2.5 | $1.3\times {10}^{-5}$ | $7.1\times {10}^{-61}$ |

**Table 6.**Represents the values of quantum tunneling probability of extracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at ${V}_{m}=0.087$ V, ${V}_{m}=0.077$ V and ${V}_{m}=0.067$ V and by setting L = 1.5 m and n = 1.

The Membrane Potential V_{m} (V) | ${\mathit{T}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{T}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

0.087 | $1.2\times {10}^{-3}$ | $8.1\times {10}^{-37}$ |

0.077 | $2.3\times {10}^{-5}$ | $1.4\times {10}^{-38}$ |

0.067 | $3.9\times {10}^{-7}$ | $2.3\times {10}^{-40}$ |

**Table 7.**Represents the values of quantum tunneling probability of extracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at n = 1, n = 2, and n = 4 and by setting ${V}_{m}=0.087$ V and $L=1.5$ m.

The Location of Gate n | ${\mathit{T}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{T}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1 | $1.2\times {10}^{-3}$ | $8.1\times {10}^{-37}$ |

2 | $4\times {10}^{-12}$ | $1.4\times {10}^{-44}$ |

4 | $1.2\times {10}^{-17}$ | $1\times {10}^{-48}$ |

**Table 8.**Represents the values of quantum tunneling probability of intracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=1$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m.

The Gate Length L (m) | ${\mathit{T}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=1\mathbf{J}$ | ${\mathit{T}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | $1.2\times {10}^{-5}$ | $5.7\times {10}^{-53}$ |

2 | $2.7\times {10}^{-7}$ | $2.2\times {10}^{-70}$ |

2.5 | $6.2\times {10}^{-9}$ | $8.5\times {10}^{-88}$ |

**Table 9.**Represents the values of quantum conductance of single channel for extracellular protons that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m and by setting ${V}_{m}=0.087$ V and n = 1.

The Gate Length L (m) | ${\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | $9.4\times {10}^{-6}$ S | $1\times {10}^{-12}$ S |

2 | $5.8\times {10}^{-6}$ S | $3\times {10}^{-15}$ S |

2.5 | $3.6\times {10}^{-6}$ S | $9\times {10}^{-18}$ S |

**Table 10.**Represents the values of quantum conductance of single channel for extracellular protons that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at ${V}_{m}=0.087$ V, ${V}_{m}=0.077$ V and ${V}_{m}=0.067$ V and by setting L = 1.5 m and n = 1.

The Membrane Potential V_{m} (V) | ${\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

0.087 | $9.4\times {10}^{-6}$ S | $1\times {10}^{-12}$ S |

0.077 | $4.3\times {10}^{-6}$ S | $4.3\times {10}^{-13}$ S |

0.067 | $1.8\times {10}^{-6}$ S | $1.8\times {10}^{-13}$ S |

**Table 11.**Represents the values of quantum conductance of single channel for extracellular protons that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at n = 1, n = 2, and n = 4 and by setting ${V}_{m}=0.087$ V and $L=1.5$ m.

The Location of Gate n | ${\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1 | $9.4\times {10}^{-6}$ S | $1\times {10}^{-12}$ S |

2 | $1.6\times {10}^{-7}$ S | $2.4\times {10}^{-14}$ S |

4 | $1.1\times {10}^{-8}$ S | $3.2\times {10}^{-15}$ S |

**Table 12.**Represents the values of quantum conductance of a single channel for intracellular protons that take the range between the two values calculated at ${E}_{Gate}=1$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m.

The Gate Length L (m) | ${\mathbf{C}}_{\mathbf{Q}}{\left(\mathbf{H}\right)}_{\mathbf{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=1\mathbf{J}$ | ${\mathbf{C}}_{\mathbf{Q}}{\left(\mathbf{H}\right)}_{\mathbf{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | $3.6\times {10}^{-6}$ S | $4.1\times {10}^{-16}$ S |

2 | $1.6\times {10}^{-6}$ S | $9.1\times {10}^{-20}$ S |

2.5 | $7.3\times {10}^{-7}$ S | $2\times {10}^{-23}$ S |

**Table 13.**Represents the values of quantum conductance of single channel for extracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m and by setting ${V}_{m}=0.087$ V and n = 1.

The Gate Length L (m) | ${\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | $4.5\times {10}^{-8}$ S | $3.2\times {10}^{-41}$ S |

2 | $4.7\times {10}^{-9}$ S | $2.9\times {10}^{-53}$ S |

2.5 | $4.9\times {10}^{-10}$ S | $2.8\times {10}^{-65}$ S |

**Table 14.**Represents the values of quantum conductance of single channel for extracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at ${V}_{m}=0.087$ V, ${V}_{m}=0.077$ V and ${V}_{m}=0.067$ V and by setting L = 1.5 m and n = 1.

The Membrane Potential V_{m} (V) | ${\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

0.087 | $4.5\times {10}^{-8}$ S | $3.2\times {10}^{-41}$ S |

0.077 | $1.1\times {10}^{-9}$ S | $5.5\times {10}^{-43}$ S |

0.067 | $1.5\times {10}^{-11}$ S | $9.1\times {10}^{-45}$ S |

**Table 15.**Represents the values of quantum conductance of single channel for extracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at n = 1, n = 2, and n = 4 and by setting ${V}_{m}=0.087$ V and $L=1.5$ m.

The Location of Gate n | ${\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | ${\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1 | $4.5\times {10}^{-8}$ S | $3.2\times {10}^{-41}$ S |

2 | $1.6\times {10}^{-16}$ S | $5.4\times {10}^{-49}$ S |

4 | $4.6\times {10}^{-22}$ S | $4\times {10}^{-53}$ S |

**Table 16.**Represents the values of quantum conductance of single channel for intracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=1$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m.

The Gate Length L (m) | ${\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=1\mathbf{J}$ | ${\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | $4.6\times {10}^{-10}$ S | $2.2\times {10}^{-57}$ S |

2 | $1.1\times {10}^{-11}$ S | $8.5\times {10}^{-75}$ S |

2.5 | $2.4\times {10}^{-13}$ S | $3.3\times {10}^{-92}$ S |

**Table 17.**Represents the values of quantum membrane conductance of extracellular protons that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m and by setting ${V}_{m}=0.087$ V, $D={10}^{11}$ channels/cm

^{2}, and n = 1.

The Gate Length L (m) | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | $9.4\times {10}^{8}$ mS/cm^{2} | 101.5 ms/cm^{2} |

2 | $5.8\times {10}^{8}$ mS/cm^{2} | 0.3 mS/cm^{2} |

2.5 | $3.6\times {10}^{8}$ mS/cm^{2} | $9\times {10}^{-4}$ mS/cm^{2} |

**Table 18.**Represents the values of quantum membrane conductance of extracellular protons that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at ${V}_{m}=0.087$ V, ${V}_{m}=0.077$ V and ${V}_{m}=0.067$ V and by setting L = 1.5 m, $D={10}^{11}$ channels/cm

^{2}, and n = 1.

The Membrane Potential V_{m} (V) | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

0.087 | $9.4\times {10}^{8}$ mS/cm^{2} | 101.5 ms/cm^{2} |

0.077 | $4.3\times {10}^{8}$ mS/cm^{2} | 43.4 mS/cm^{2} |

0.067 | $1.8\times {10}^{8}$ mS/cm^{2} | $18.3$ mS/cm^{2} |

**Table 19.**Represents the values of quantum membrane conductance of extracellular protons that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at n = 1, n = 2, and n = 4 and by setting ${V}_{m}=0.087$ V, $L=1.5$ m, and $D={10}^{11}$ channels/cm

^{2}.

The Location of Gate n | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1 | $9.4\times {10}^{8}$ mS/cm^{2} | 101.5 ms/cm^{2} |

2 | $1.6\times {10}^{7}$ mS/cm^{2} | 2.4 mS/cm^{2} |

4 | $1.1\times {10}^{6}$ mS/cm^{2} | 0.32 mS/cm^{2} |

**Table 20.**Represents the values of quantum membrane conductance of extracellular protons that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at $D={10}^{11}$ channels/cm

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}and by setting ${V}_{m}=0.087$ V, $L=1.5$ m, and n = 1.

The Density of Channels D (Channels/cm ^{2}) | ||
---|---|---|

${10}^{11}$ | $9.4\times {10}^{8}$ mS/cm^{2} | 101.5 ms/cm^{2} |

${10}^{10}$ | $9.4\times {10}^{7}$ mS/cm^{2} | 10.2 mS/cm^{2} |

${10}^{9}$ | $9.4\times {10}^{6}$ mS/cm^{2} | 1.01 mS/cm^{2} |

**Table 21.**Represents the values of quantum membrane conductance of intracellular protons that take the range between the two values calculated at ${E}_{Gate}=1$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m and by setting $D={10}^{11}$ channels/cm

^{2}.

The Gate Length L (m) | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=1\mathbf{J}$ | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | $3.6\times {10}^{8}$ mS/cm^{2} | 0.041 ms/cm^{2} |

2 | $1.6\times {10}^{8}$ mS/cm^{2} | $9.1\times {10}^{-6}$ mS/cm^{2} |

2.5 | $7.3\times {10}^{7}$ mS/cm^{2} | $2\times {10}^{-9}$ mS/cm^{2} |

**Table 22.**Represents the values of quantum membrane conductance of intracellular protons that take the range between the two values calculated at ${E}_{Gate}=1$ J and ${E}_{Gate}=7$ J. The evaluation is made at $D={10}^{11}$ channels/cm

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}, and by setting $L=1.5$ m.

The Density of Channels D (Channels/cm ^{2}) | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=1\mathbf{J}$ | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{H})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

${10}^{11}$ | $3.6\times {10}^{8}$ mS/cm^{2} | 0.041 ms/cm^{2} |

${10}^{10}$ | $3.6\times {10}^{7}$ mS/cm^{2} | $4.1\times {10}^{-3}$ mS/cm^{2} |

${10}^{9}$ | $3.6\times {10}^{6}$ mS/cm^{2} | $4.1\times {10}^{-4}$ mS/cm^{2} |

**Table 23.**Represents the values of quantum membrane conductance of extracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m and by setting ${V}_{m}=0.087$ V, $D={10}^{11}$ channels/cm

^{2}, and n = 1.

The Gate Length L (m) | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | $4.5\times {10}^{6}$ mS/cm^{2} | $3.2\times {10}^{-27}$ ms/cm^{2} |

2 | $4.7\times {10}^{5}$ mS/cm^{2} | $2.9\times {10}^{-39}$ mS/cm^{2} |

2.5 | $4.9\times {10}^{4}$ mS/cm^{2} | $2.8\times {10}^{-51}$ mS/cm^{2} |

**Table 24.**Represents the values of quantum membrane conductance of extracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at ${V}_{m}=0.087$ V, ${V}_{m}=0.077$ V and ${V}_{m}=0.067$ V and by setting L = 1.5 m, $D={10}^{11}$ channels/cm

^{2}, and n = 1.

The Membrane Potential V_{m} (V) | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

0.087 | $4.5\times {10}^{6}$ mS/cm^{2} | $3.2\times {10}^{-27}$ ms/cm^{2} |

0.077 | $1.1\times {10}^{5}$ mS/cm^{2} | $5.5\times {10}^{-29}$ mS/cm^{2} |

0.067 | $1.5\times {10}^{3}$ mS/cm^{2} | $9.1\times {10}^{-31}$ mS/cm^{2} |

**Table 25.**Represents the values of quantum membrane conductance of extracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at n = 1, n = 2, and n = 4 and by setting ${V}_{m}=0.087$ V, $L=1.5$ m, and $D={10}^{11}$ channels/cm

^{2}.

The Location of Gate n | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=2.5\mathbf{J}$ | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{E}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1 | $4.5\times {10}^{6}$ mS/cm^{2} | $3.2\times {10}^{-27}$ ms/cm^{2} |

2 | $1.6\times {10}^{-2}$ mS/cm^{2} | $5.4\times {10}^{-35}$ mS/cm^{2} |

4 | $4.6\times {10}^{-8}$ mS/cm^{2} | $4\times {10}^{-39}$ mS/cm^{2} |

**Table 26.**Represents the values of quantum membrane conductance of extracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=2.5$ J and ${E}_{Gate}=7$ J. The evaluation is made at $D={10}^{11}$ channels/cm

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}and by setting ${V}_{m}=0.087$ V, $L=1.5$ m, and n = 1.

The Density of Channels D (Channels/cm ^{2}) | ||
---|---|---|

${10}^{11}$ | $4.5\times {10}^{6}$ mS/cm^{2} | $3.2\times {10}^{-27}$ ms/cm^{2} |

${10}^{10}$ | $4.5\times {10}^{5}$ mS/cm^{2} | $3.2\times {10}^{-28}$ mS/cm^{2} |

${10}^{9}$ | $4.5\times {10}^{4}$ mS/cm^{2} | $3.2\times {10}^{-29}$ mS/cm^{2} |

**Table 27.**Represents the values of quantum membrane conductance of intracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=1$ J and ${E}_{Gate}=7$ J. The evaluation is made at L = 1.5 m, L = 2 m, and L = 2.5 m and by setting $D={10}^{11}$ channels/cm

^{2}.

The Gate Length L (m) | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=1\mathbf{J}$ | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

1.5 | $4.6\times {10}^{4}$ mS/cm^{2} | $2.2\times {10}^{-43}$ ms/cm^{2} |

2 | $1.1\times {10}^{3}$ mS/cm^{2} | $8.5\times {10}^{-61}$ mS/cm^{2} |

2.5 | $24$ mS/cm^{2} | $3.3\times {10}^{-78}$ mS/cm^{2} |

**Table 28.**Represents the values of quantum membrane conductance of intracellular sodium ions that take the range between the two values calculated at ${E}_{Gate}=1$ J and ${E}_{Gate}=7$ J. The evaluation is made at $D={10}^{11}$ channels/cm

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}, and by setting $L=1.5$ m.

The Density of Channels D (Channels/cm ^{2}) | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=1\mathbf{J}$ | $\mathit{M}{\mathit{C}}_{\mathit{Q}}{(\mathit{N}\mathit{a})}_{\mathit{I}}$$\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=7\mathbf{J}$ |
---|---|---|

${10}^{11}$ | $4.6\times {10}^{4}$ mS/cm^{2} | $2.2\times {10}^{-43}$ ms/cm^{2} |

${10}^{10}$ | $4.6\times {10}^{3}$ mS/cm^{2} | $2.2\times {10}^{-44}$ mS/cm^{2} |

${10}^{9}$ | $4.6\times {10}^{2}$ mS/cm^{2} | $2.2\times {10}^{-45}$ mS/cm^{2} |

**Table 29.**Represents the values of point of curving for protons ${E}_{cur}(H)$, the membrane potential at ${E}_{Gate}=1$ J, and the average rate of depolarization. The evaluation is made at three different values of $p{H}_{E}$ ($p{H}_{E}=7.4$, $p{H}_{E}=7$, and $p{H}_{E}=6.5$) and by setting $L=1.5$ m, $D={10}^{11}$ channels/cm

^{2}, and n = 1.

$\mathit{p}{\mathit{H}}_{\mathit{E}}$ | $\mathbf{Point}\mathbf{of}\mathbf{Curving}{\mathit{E}}_{\mathit{c}\mathit{u}\mathit{r}}(\mathit{H})\left(\mathbf{J}\right)$ | $\mathbf{Membrane}\mathbf{Potential}\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=1\mathbf{J}$ | $\mathbf{Average}\mathbf{Rate}\mathbf{of}\mathbf{Depolarization}\mathit{R}(\mathit{H})(\mathbf{V}/\mathbf{J})$ |
---|---|---|---|

7.4 | 5.92 | 0.014 V | $1.5\times {10}^{-2}$ |

7 | 6.22 | 0.014 V | $1.4\times {10}^{-2}$ |

6.5 | 6.61 | 0.014 V | $1.3\times {10}^{-2}$ |

**Table 30.**Represents the values of point of curving for protons ${E}_{cur}(H)$, the membrane potential at ${E}_{Gate}=1$ J, and the average rate of depolarization. The evaluation is made at three different values of gate length ($L=1.5$ m, $L=2$ m, and $L=2.5$ m) and by setting $p{H}_{E}=7.4$, $D={10}^{11}$ channels/cm

^{2}, and n = 1.

The Gate Length L (m) | $\mathbf{Point}\mathbf{of}\mathbf{Curving}{\mathit{E}}_{\mathit{c}\mathit{u}\mathit{r}}(\mathit{H})\left(\mathbf{J}\right)$ | $\mathbf{Membrane}\mathbf{Potential}\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=1\mathbf{J}$ | $\mathbf{Average}\mathbf{Rate}\mathbf{of}\mathbf{Depolarization}\mathit{R}(\mathit{H})(\mathbf{V}/\mathbf{J})$ |
---|---|---|---|

1.5 | 5.92 | 0.014 V | $1.5\times {10}^{-2}$ |

2 | 4.83 | 0.01 V | $2\times {10}^{-2}$ |

2.5 | 4.24 | 0.0084 V | $2.4\times {10}^{-2}$ |

**Table 31.**Represents the values of point of curving for protons ${E}_{cur}(H)$, the membrane potential at ${E}_{Gate}=1$ J, and the average rate of depolarization. The evaluation is made at three different values of channels density ($D={10}^{11}$ channels/cm

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}) and by setting $p{H}_{E}=7.4$, $L=1.5$ m, and n = 1.

The Density of Channels D (Channels/cm^{2}) | $\mathbf{Point}\mathbf{of}\mathbf{Curving}{\mathit{E}}_{\mathit{c}\mathit{u}\mathit{r}}(\mathit{H})\left(\mathbf{J}\right)$ | $\mathbf{Membrane}\mathbf{Potential}\mathbf{at}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}=1\mathbf{J}$ | $\mathbf{Average}\mathbf{Rate}\mathbf{of}\mathbf{Depolarization}\mathit{R}(\mathit{H})(\mathbf{V}/\mathbf{J})$ |
---|---|---|---|

${10}^{11}$ | 5.92 | 0.014 V | $1.5\times {10}^{-2}$ |

${10}^{10}$ | 5.2 | 0.014 V | $1.7\times {10}^{-2}$ |

${10}^{9}$ | 4.54 | 0.014 V | $2\times {10}^{-2}$ |

**Table 32.**Represents the values of point of curving for protons ${E}_{cur}(H)$, the membrane potential at ${E}_{Gate}=1$ J, and the average rate of depolarization. The evaluation is made at three different values of gate location (n = 1, n = 2, and n = 4) and by setting $p{H}_{E}=7.4$, $L=1.5$ m, and $D={10}^{11}$ channels/cm

^{2}.

The Location of Gate n | |||
---|---|---|---|

1 | 5.92 | 0.014 V | $1.5\times {10}^{-2}$ |

2 | 4.65 | 0.022 V | $1.8\times {10}^{-2}$ |

4 | 3.97 | 0.031 V | $1.9\times {10}^{-2}$ |

**Table 33.**Represents the values of point of curving for sodium ions ${E}_{cur}(Na)$, the zero membrane potential and the corresponding ${E}_{Gate}$, and the average rate of depolarization. The evaluation is made at three different values of gate length ($L=1.5$ m, $L=2$ m, and $L=2.5$ m) and by setting $D={10}^{11}$ channels/cm

^{2}, and n = 1.

The Gate Length L (m) | $\mathbf{Point}\mathbf{of}\mathbf{Curving}{\mathit{E}}_{\mathit{c}\mathit{u}\mathit{r}}(\mathit{N}\mathit{a})\left(\mathbf{J}\right)$ | $\mathbf{Zero}\mathbf{Membrane}\mathbf{Potential}\mathbf{and}\mathbf{the}\mathbf{Corresponding}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}$ | $\mathbf{Average}\mathbf{Rate}\mathbf{of}\mathbf{Depolarization}\mathit{R}(\mathit{N}\mathit{a})(\mathbf{V}/\mathbf{J})$ |
---|---|---|---|

1.5 | 3.62 | 0 V at ${E}_{Gate}=1.33$ J | $3.8\times {10}^{-2}$ |

2 | 3.25 | 0 V at ${E}_{Gate}=1.16$ J | $4.1\times {10}^{-2}$ |

2.5 | 3.03 | 0 V at ${E}_{Gate}=1.06$ J | $4.4\times {10}^{-2}$ |

**Table 34.**Represents the values of point of curving for sodium ions ${E}_{cur}(Na)$, the zero membrane potential and the corresponding ${E}_{Gate}$, and the average rate of depolarization. The evaluation is made at three different values of channels density ($D={10}^{11}$ channels/cm

^{2}, $D={10}^{10}$ channels/cm

^{2}, and $D={10}^{9}$ channels/cm

^{2}) and by setting $L=1.5$ m, and n = 1.

The Density of Channels D (Channels/cm^{2}) | $\mathbf{Point}\mathbf{of}\mathbf{Curving}{\mathit{E}}_{\mathit{c}\mathit{u}\mathit{r}}(\mathit{N}\mathit{a})\left(\mathbf{J}\right)$ | $\mathbf{Zero}\mathbf{Membrane}\mathbf{Potential}\mathbf{and}\mathbf{the}\mathbf{Corresponding}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}$ | $\mathbf{Average}\mathbf{Rate}\mathbf{of}\mathbf{Depolarization}\mathit{R}(\mathit{N}\mathit{a})(\mathbf{V}/\mathbf{J})$ |
---|---|---|---|

${10}^{11}$ | 3.62 | 0 V at ${E}_{Gate}=1.33$ J | $3.76\times {10}^{-2}$ |

${10}^{10}$ | 3.5 | 0 V at ${E}_{Gate}=1.26$ J | $3.84\times {10}^{-2}$ |

${10}^{9}$ | 3.38 | 0 V at ${E}_{Gate}=1.19$ J | $3.9\times {10}^{-2}$ |

**Table 35.**Represents the values of point of curving for sodium ions ${E}_{cur}(Na)$, the zero membrane potential and the corresponding ${E}_{Gate}$, and the average rate of depolarization. The evaluation is made at three different values of gate location (n= 1, n = 2, and n = 4) and by setting $L=1.5$ m, and $D={10}^{11}$ channels/cm

^{2}.

The Location of Gate n | $\mathbf{Point}\mathbf{of}\mathbf{Curving}{\mathit{E}}_{\mathit{c}\mathit{u}\mathit{r}}(\mathit{N}\mathit{a})\left(\mathbf{J}\right)$ | $\mathbf{Zero}\mathbf{Membrane}\mathbf{Potential}\mathbf{and}\mathbf{the}\mathbf{Corresponding}{\mathit{E}}_{\mathit{G}\mathit{a}\mathit{t}\mathit{e}}$ | $\mathbf{Average}\mathbf{Rate}\mathbf{of}\mathbf{Depolarization}\mathit{R}(\mathit{N}\mathit{a})(\mathbf{V}/\mathbf{J})$ |
---|---|---|---|

1 | 3.62 | 0 V at ${E}_{Gate}=1.33$ J | $3.8\times {10}^{-2}$ |

2 | 2.62 | 0 V at ${E}_{Gate}=1.33$ J | $6.7\times {10}^{-2}$ |

4 | 2.1 | 0 V at ${E}_{Gate}=1.33$ J | $0.11$ |

**Table 36.**Represents the values of membrane potential that take the range between the two values calculated at $p{H}_{E}=7.4$ and at $p{H}_{E}=5$ under the influence of classical transport through open sodium channels. The evaluation is made at three different setting values of channels density ($D={10}^{11}$ channels/cm

^{2}, $D={10}^{10}$ channels/cm

^{2}, $D={10}^{9}$ channels/cm

^{2}) and by setting the permeability ratio ${P}_{H}/{P}_{Na}=252$, and the conductance of single channel ${C}_{Na}=17.3\times {10}^{-12}$ S.

The Density of Channels (Channels/cm^{2}) | $\mathbf{Membrane}\mathbf{Potential}\left(\mathbf{V}\right)\mathbf{at}\mathit{p}{\mathit{H}}_{\mathit{E}}=7.4$ | $\mathbf{Membrane}\mathbf{Potential}\left(\mathbf{V}\right)\mathbf{at}\mathit{p}{\mathit{H}}_{\mathit{E}}=5$ |
---|---|---|

${10}^{11}$ | 0.067 | 0.062 |

${10}^{10}$ | 0.08 | 0.062 |

${10}^{9}$ | 0.086 | 0.064 |

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**MDPI and ACS Style**

Ababneh, O.; Qaswal, A.B.; Alelaumi, A.; Khreesha, L.; Almomani, M.; Khrais, M.; Khrais, O.; Suleihat, A.; Mutleq, S.; Al-olaimat, Y.;
et al. Proton Quantum Tunneling: Influence and Relevance to Acidosis-Induced Cardiac Arrhythmias/Cardiac Arrest. *Pathophysiology* **2021**, *28*, 400-436.
https://doi.org/10.3390/pathophysiology28030027

**AMA Style**

Ababneh O, Qaswal AB, Alelaumi A, Khreesha L, Almomani M, Khrais M, Khrais O, Suleihat A, Mutleq S, Al-olaimat Y,
et al. Proton Quantum Tunneling: Influence and Relevance to Acidosis-Induced Cardiac Arrhythmias/Cardiac Arrest. *Pathophysiology*. 2021; 28(3):400-436.
https://doi.org/10.3390/pathophysiology28030027

**Chicago/Turabian Style**

Ababneh, Omar, Abdallah Barjas Qaswal, Ahmad Alelaumi, Lubna Khreesha, Mujahed Almomani, Majdi Khrais, Oweiss Khrais, Ahmad Suleihat, Shahed Mutleq, Yazan Al-olaimat,
and et al. 2021. "Proton Quantum Tunneling: Influence and Relevance to Acidosis-Induced Cardiac Arrhythmias/Cardiac Arrest" *Pathophysiology* 28, no. 3: 400-436.
https://doi.org/10.3390/pathophysiology28030027